A Four-Point Theorem: Yet Another Variation on an Old Theme

The subject of this article belongs to a “neighborhood” of the four-vertex theorem, which in its simplest form, states that the curvature of a plane oval (a smooth closed curve with positive curvature) has at least four critical points. Since its publication by Syamadas Mukhopadhyaya in 1909, this result and its ramifications have generated a vast literature. We give but one reference: [5, Lecture 10]. In what follows, we freely use basic facts of elementary differential geometry of the sphere and the hyperbolic plane, and we omit references to numerous textbooks on the subject.